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Binary Protocol Processing same as for ternary protocol Binary signaling – takes one of two states 0 or 1 e e signals 0 or 1 with equal probability 14

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Binary Signaling – A Motivation Nodes may not be able to signal indifference – by the very nature of the application Ex. two news pieces may be equally most read but only one can be recommended to the user 15 US navy ship stems into port where Russian... Soldier forced to sleep in car after hotel...

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Questions of Interest Probability of convergence to incorrect consensus ? Time to reach consensus ? Dependence on the number of nodes N and initial fraction of nodes holding the majority state ? 16

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This Talk Assumptions Complete graph node interactions Each node samples a node uniformly at random across all nodes at instances of a Poisson process with intensity 1 Arbitrary graph interactions of interest – for future work 17

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Summary of Results – Talk Outline Ternary protocol Prob of error decays exponentially with the number of nodes N – found exact exponent log(N) convergence time Binary protocol Prob of error worse than for ternary protocol for a factor exponentially increasing with N, but not worse than for classical voter Convergence time C log(N) with 2 C 3 18

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Ternary Protocol - Dynamics U = number of nodes in state 0 V = number of nodes in state 1 N = total number of nodes 19 (U,V) Markov process:

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Conclusion Good news results for binary consensus on complete graphs Ternary signaling Probability of error decays exponentially with the number of nodes N log(N) convergence time Binary signaling Probability of error worse than for the ternary signaling for a factor exponentially increasing with N, but not worse than for classical voter Convergence time C log(N) with 2 C 3 36